core signalling motif displaying multistability through multi-state...
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ResearchCite this article: Feng S, Saez M, Wiuf C,
Feliu E, Soyer OS. 2016 Core signalling motif
displaying multistability through multi-state
enzymes. J. R. Soc. Interface 13: 20160524.
http://dx.doi.org/10.1098/rsif.2016.0524
Received: 1 July 2016
Accepted: 6 September 2016
Subject Category:Life Sciences – Mathematics interface
Subject Areas:biochemistry, systems biology,
synthetic biology
Keywords:bistability, futile cycles, synthetic biology,
signalling networks, competition
Authors for correspondence:Elisenda Feliu
e-mail: [email protected]
Orkun S. Soyer
e-mail: [email protected]
†These authors contributed equally to this
study.
Electronic supplementary material is available
online at https://dx.doi.org/10.6084/m9.
figshare.c.3491544.
& 2016 The Authors. Published by the Royal Society under the terms of the Creative Commons AttributionLicense http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the originalauthor and source are credited.Core signalling motif displayingmultistability through multi-stateenzymes
Song Feng1,†, Meritxell Saez2,†, Carsten Wiuf2, Elisenda Feliu2
and Orkun S. Soyer1
1School of Life Sciences, University of Warwick, Coventry, UK2Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen,Denmark
OSS, 0000-0002-9504-3796
Bistability, and more generally multistability, is a key system dynamics feature
enabling decision-making and memory in cells. Deciphering the molecular
determinants of multistability is thus crucial for a better understanding of cel-
lular pathways and their (re)engineering in synthetic biology. Here, we show
that a key motif found predominantly in eukaryotic signalling systems,
namely a futile signalling cycle, can display bistability when featuring a
two-state kinase. We provide necessary and sufficient mathematical con-
ditions on the kinetic parameters of this motif that guarantee the existence
of multiple steady states. These conditions foster the intuition that bistability
arises as a consequence of competition between the two states of the kinase.
Extending from this result, we find that increasing the number of kinase
states linearly translates into an increase in the number of steady states in
the system. These findings reveal, to our knowledge, a new mechanism for
the generation of bistability and multistability in cellular signalling systems.
Further the futile cycle featuring a two-state kinase is among the smallest bis-
table signalling motifs. We show that multi-state kinases and the described
competition-based motif are part of several natural signalling systems and
thereby could enable them to implement complex information processing
through multistability. These results indicate that multi-state kinases in signal-
ling systems are readily exploited by natural evolution and could equally be
used by synthetic approaches for the generation of multistable information
processing systems at the cellular level.
1. IntroductionCells sense environmental stimuli and use these to initiate appropriate physio-
logical responses. Understanding such cellular information processing in
healthy and diseased states [1–3], and engineering it through synthetic biology
[4–7], requires better insights into the relationship between different interaction
motifs found in signalling networks and their potential roles in the ensuing
system dynamics [8]. To this end, a key interaction motif found predominantly
in eukaryotic signalling systems is that of a futile signalling cycle, where a sub-
strate protein is phosphorylated by a kinase and dephosphorylated by a
phosphatase. When these enzymes are saturated by their substrate, this motif
can display ultrasensitive response dynamics, enabling threshold responses to
graded input signals [9]. It can also be shown theoretically, that the futile
cycle motif in its simple form cannot enable bistability (see below). Experimen-
tal studies of cellular systems embedding the futile signalling cycle for several
physiological responses, including cell fate determination and cell division
[10–13], found ultrasensitive responses and in some cases bistability [14–21].
While the presence of bistability has been indicated to be functionally
structure
Sp
S
Sp
S
K
Kr
Kt
Sp
S
Kr
Kt
P
P
reactions allow bistability
no
yes
yesSp S
K + S ↽⇀ KS K + Sp
P + Sp ↽⇀ PSp P + S
Kr + S ↽⇀ KrS Kr + Sp
Kt + S ↽⇀ KtS Kt + Sp
P + Sp ↽⇀ PSp P + S
Kr ↽⇀ Kt KrS ↽⇀ KtS
Kr + S ↽⇀ KrS Kr + Sp
Kt + S ↽⇀ KtS Kt + Sp
Kr ↽⇀ Kt KrS ↽
⇀ KtS
(a)
(b)
(c)
↽⇀↽⇀
Figure 1. Cartoon of three signalling motifs. (a). A simple futile signallingcycle with a substrate that has a single phosphorylation site and is actedupon by a kinase and phosphatase. Both enzymes are assumed not to beallosteric. (b). The proposed bistable futile signalling cycle derived from(a). The substrate is phosphorylated by a two-state kinase and dephosphory-lated by a single-state phosphatase. (c). The core signalling motif consistingof a futile cycle with a two-state kinase. The phosphorylated substrate under-goes auto-dephosphorylation at a constant rate.
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significant, for example, in the generation of phenotypic
variability [22–25], its molecular implementations have not
been fully elucidated.
To achieve bistability in a futile signalling cycle motif,
the originally studied structure of this motif needs to be
extended with additional features. Theoretical studies have
shown that bistability can be achieved if there are feedback
interactions between the substrate and its acting enyzmes
(i.e. the kinase or phosphatase) [26–30], or if the substrate
has multiple phosphorylation sites [31–34]. The latter prop-
osition is particularly interesting as the presence of multiple
phosphorylation sites on signalling proteins is a common
phenomenon [35,36]. Kinases, phosphatases, as well as their
substrates readily exhibit two or more conformational states
that are associated with different levels of phosphorylation
and that result in different catalytic activity levels [37–39].
In the signalling pathways regulating the cell cycle,
for example, it has been hypothesized that signalling pro-
teins with multiple phosphorylation sites act as multi-state
enzymes that can embed complex signal processing
[38–41]. It is also shown that the different activity levels
of signalling proteins can be regulated through allosteric
interactions with ligands or other proteins, such as the so-
called scaffolding proteins [37,42–44]. Scaffolding proteins,
which are ubiquitous in signalling systems [42,43], can also
have multiple phosphorylation and binding sites themselves
and, as such, are key regulators in signalling pathways
[11,45–48]. Despite these experimental findings and obser-
vations on specific signalling proteins and pathways, it has
been difficult to elucidate any particular features, or design
principles, that can provide a clear understanding between
the nature of signal processing that a system implements
and the presence of multi-phosphorylation-site-featuring
multi-state enzymes. This difficulty arises partially from the
fact that modelling of signalling pathways with multi-state
enzymes becomes increasingly complex, with a combinatorial
explosion of possible interactions in the system.
Here, we perform a systematic, mathematical analysis of
the effects of having multi-state kinases on the response
dynamics and the number of steady states in a simple and
core futile signalling cycle motif. When this motif is analysed
with the assumption of single-state enzymes, the resulting
system cannot display bistability for any positive kinetic par-
ameter values. This situation changes and bistability becomes
possible with the introduction of a two-state kinase, leading
to one of the smallest signalling systems that is bistable and
comparable in size to previously identified minimal systems
[12,22,31–34,49–54]. Using this minimalist system as a tract-
able core motif, we are able to derive mathematical
conditions on the kinetic parameters and/or the total concen-
trations of substrate and kinase that are necessary and
sufficient for the existence of three steady states. This allows
an intuitive insight that bistability in this minimalist system
arises from the competition between the different states of
the kinase for the substrate. Extending from this intuition,
we show that increasing the number of kinase states in the
system leads to a linear increase in the number of steady
states. We show that both multi-state enzymes and the dis-
cussed core motif are prevalent in many signalling
pathways and that the identified parameter ranges for bi-
stability are biologically plausible. These results provide an
intuitive view on multi-state enzymes leading to bistability
and multistability through competition for their substrates.
As such, the multi-state nature of enzymes can be exploited
to better understand natural signalling pathways and to
engineer novel ones.
2. Results2.1. The futile signalling cycle with a two-state kinase
is a bistable motifA key interaction motif found in eukaryotic signalling networks
is the so-called futile signalling cycle (figure 1a). When con-
sidered with a single phosphorylation site on the substrate
and a simple, one-state kinase and phosphatase, this motif
cannot display bistability for any choice of positive parameters
(e.g. [34,55]). When we extended this system with a two-state
kinase, this key result changed and bistability was possible.
We introduced the two-state kinase such that each state can
bind the substrate and catalyse its phosphorylation, and
where transitions between the two states are possible irrespec-
tive of substrate binding (figure 1b). The two-state kinase in
this simple model switches between two conformational
states with a constant rate. The two states show differential cat-
alytic activity towards the substrate (figure 1b, Material and
methods). While this is the simplest model to introduce the
idea of multi-state enzymes into the core futile cycle motif, it
is readily possible to assume more complex models. In particu-
lar, the conformational change between kinase states can be
modelled as an allosteric regulation [56–60], whereby it is
linked to binding of the kinases by a ligand or other proteins,
or as arising from covalent phosphorylation events as com-
monly observed in signalling proteins [26,36,61]. We consider
such complex models below, but note that adding this complex-
ity does not alter the key conclusions of this study on bistability
and multistability.
We find that the core motif with two-state kinases can
be further simplified without compromising bistability by
removing the phosphatase and letting the dephosphorylation
of the substrate happen through auto-hydrolysis at a constant
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rate (figure 1c, Material and methods). In this way, we obtain a
minimal core signalling system driven by a two-state kinase,
which displays bistability. The system contains only six species,
making it one of the smallest bistable signalling motifs.
2.2. Conditions for bistability in the core motif aresatisfied in a biologically plausible range
The simplicity of this core motif allowed us to analytically
study the solutions to the steady-state equations (see the elec-
tronic supplementary material). In particular, we were able to
derive a set of inequalities in the kinetic parameters and total
concentrations of the substrate and kinase that provide a set
of necessary and sufficient conditions for the existence of
three steady states in the system (equation (4.5), see also
the electronic supplementary material for the derivation of
this equation). While equation (4.5) provides a necessary
and sufficient condition on parameters for the presence
of multistationarity, we also derive a simpler equation
representing the necessary condition for bistability:
(k3 � k6)(hrk9k10 � htk8k11) . 0, ð2:1Þ
where the indexing of the rate constants is as given in equation
(4.1) in Material and methods, and the composite parameters
hr ¼ k1=ðk2 þ k3Þ and ht ¼ k4=ðk5 þ k6Þ are the inverses of
the Michaelis–Menten constants of Kr (the kinase at the relaxed
state) and Kt (the kinase at the tense state), respectively. Analy-
sis of this equation reveals some of the key structural features of
the system that are necessary for bistability, in particular: (i) the
conversion between the two free forms of the kinase and
(ii) the conversion between the two substrate-bound forms of
the kinase. That is, both k8 and k9 cannot be zero, and both
k10 and k11 cannot be zero. Thus, the structure of the reaction
system comprised of a futile signalling cycle driven by a
two-state kinase is crucial for enabling bistability.
Furthermore, equation (2.1) provides two key dynamical
features for bistability. Firstly, the two interconnected futile
cycles between S and Sp, defined by the two kinase states,
need to operate at different catalytic rates (i.e. k3 = k6). Sec-
ondly, the switching between these cycles through the four
forms of the kinase (i.e. Kr, Kt, KrS, KtS) needs to occur at
different rates, and in a way opposing the difference in the
catalytic rates. Specifically, if the futile cycle for the relaxed
state of the kinase (i.e. Kr and KrS) has the highest catalytic
activity (i.e. k3 . k6), then hrk9k10 needs to be larger than
htk8k11. As a consequence, the clockwise interchanging
cycle, Kr ! KrS! KtS! Kt ! Kr, corresponding to the pro-
duct of the rate constants k1k10(k5 þ k6)k9, needs to dominate
over the anti-clockwise cycle, Kt ! KtS! KrS! Kr ! Kt,
corresponding to the product k4k11(k2 þ k3)k8. Symmetri-
cally, if Kt has higher catalytic activity than Kr (i.e. k3 , k6),
then the anti-clockwise cycle needs to dominate (see also
figure 2 and discussion below).
A further constraint on the rates governing the transitions
among the four forms of the kinase might arise from thermo-
dynamics. Particularly, these transitions form a local reaction
cycle, which must follow the principle of detailed balance if we
assume no additional energy input into the system [62–65].
This results in a thermodynamic constraint on the reaction
kinetics such that the product of the rate constants in the clock-
wise direction must equal the product of the reverse rate
constants (i.e. k1k5k9k10¼ k2k4k8k11). It must also be noted,
however, that this constraint would be relaxed if the
conformational switching between the enzyme states were
directed by energy input (e.g. phosphorylation–dephosphoryla-
tion reactions; electronic supplementary material, figure S1a) or
steric effects with enzyme binding with other proteins or
enzymes (electronic supplementary material, figure S1b).
To determine whether these conditions on kinetic rates
can be simultaneously satisfied in cellular signalling net-
works, we tabulated kinetic parameters from the literature
(see the electronic supplementary material, table S1 and refer-
ences therein). We then sampled 105 parameter sets around
these known kinetic parameters and checked whether the
necessary and sufficient conditions for bistability were satis-
fied (see Material and methods). This analysis showed that
the futile signalling cycle displays bistability in a biologically
plausible parameter regime, even when thermodynamic con-
straints are taken into account (electronic supplementary
material, figure S2 and table S2).
2.3. Bistability can be seen as arising from competitionbetween the kinase states for the substrate andresulting positive feedback loops
It is interesting to note that the mathematical conditions
derived in equation (2.1) impose a specific structure onto
the core motif. As discussed above, this comprises the conver-
sion between the four, free and substrate-bound forms of the
kinase. The interactions of these different kinase states with
the substrate can be seen as two connected reaction cycles
involving competition for the same substrate (figure 2a).
Equation (2.1) shows that the flows of these two competing
reaction cycles need to have a specific relationship for bist-
ability to emerge. To better understand these ensuing
reaction fluxes, we have analysed the steady states of the
system for increasing total kinase concentration, as a proxy
for an increasing signal (figure 2b, see also the electronic sup-
plementary material, sections 1.8 and 1.9). For a fixed set of
parameters in the bistable regime such that k3 . k6, k9 . k8,
k10 . k11 and hrk9k10 . htk8k11 (see the electronic sup-
plementary material, table S3), we find that in the low
signal regime, where the total level of kinase is low, there is
a large flux from KrS into KtS, resulting in the accumulation
of KtS. Thus in this low signal regime, the slow futile cycle
driven by Kt (which has the lowest catalytic activity) domi-
nates (i.e. [Kr] þ [KrS] , [Kt] þ [KtS]) and the system is at
‘low’ state (i.e. small [Sp]) (figure 2b, red dots). In the high
signal regime, the fast futile cycle driven by Kr dominates
(i.e. [Kr] þ [KrS] . [Kt] þ [KtS]) and the system is at the
‘high’ state (i.e. large [Sp]). The substrate is largely conver-
ted to the phosphorylated form, which results in the
accumulation of Kr (figure 2b, green dots). Whether the
Kr-mediated or Kt-mediated cycle dominates is determined by
the interplay between the catalytic constants, the Michaelis–
Menten constants associated with each kinase form, and the
transition rate constants between these forms in a free and
substrate-bound state.
This analysis derived from the necessary parameter con-
dition (equation (2.1)) leads to an intuitive view, in which
the bistability in the system is understood as a result of the
two futile cycles driven by the two forms of the kinase com-
peting for the substrate. Furthermore, the competing kinase
forms need to have opposite dominance in terms of being
able to bind the substrate and their catalytic activity, such
Kr KrS
KtSKt
S Sp
k1
k3k2
k11
k10
k6
k9
k8
k5
k4
k7
[Ktot] (mM)
[Sp]
(mM
)
10
8
6
4
2
0 0.5 1.0 1.5 2.0 2.5 3.0
(a) (b)
Figure 2. Schematic of the core signalling motif displaying bistability. (a) Cartoon representation of the two interconnected reaction cycles constituting the corebistable system. The arrows represent reactions in the system and are labelled with the kinetic parameters from equation (4.2). Two rectangles (dashed line)with different background colour show the two futile cycles with Kr (green) and Kt (red) competing for the substrate (in the intersected region of the tworectangles). (b) Bifurcation plot of core bistable signalling motif. The solid line corresponds to the stable steady-state levels of [Sp] with increasing signalgiven by the total concentration of kinase [Ktot]. The dashed line corresponds to the unstable steady states. The parameter values used to generate the bifurcationplot are listed in the electronic supplementary material, table S3. The four little cartoons, drawn as inset, are showing the allocation of all species’ concentrationsand corresponding reaction fluxes at the different levels of [Ktot], as indicated by the coloured dots. For each cartoon, the size of the species’ boxes and reactionarrows are calculated from the actual species’ concentrations and the levels of the reaction fluxes, namely k1[KrS], k2[KrS], k3[KrS], k4[KtS], k5[KtS], k6[KtS],k7[Sp], k8[Kr], k9[Kt], k10[KrS] and k11[KtS]. Specifically, the species’ concentrations and reaction flux values are mapped on the cartoon by log-transformingthe actual value and then mapping these onto a specific interval; the intervals used for arrow thickness and box size were [0.5,13] and [20,100], respectively.The transparency index for the two overlay boxes indicating two futile cycles mediated by Kr (green) and Kt (red) are calculated by log-transforming the totalconcentrations of Kr and Kt containing species and mapping these onto an interval [0,100].
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that the form dominating catalytically needs to be weaker
in terms of substrate binding kinetics (after correction by
the transition rate constants). For example, if k3 . k6, then
we need to have hrk9k10 . htk8k11. Interestingly, this motif
structure embedding competition between the kinase forms
for the substrate also gives rise to positive feedback loops
[66] that are not readily seen from the reaction cartoon, but
can be found within the bipartite graph that encodes how
the species and reactions of the network influence each
other (see the electronic supplementary material, SI 1.10).
2.4. Increasing the number of kinase states in the futilecycle increases the number of steady states(theoretically unbounded multistationarity)
Recognizing that bistability in the core motif is linked to the
competition between the two futile cycles, it is intriguing to
consider whether adding more competing cycles increases
the number of steady states. To expand from the simplest
motif towards more complicated systems, one way of increas-
ing competing cycles is to increase the number of two-state
kinases, while the other is to increase the number of states of
a single kinase. We find that both expansions of the minimal
system result in an increase of the number of steady states.
Firstly, we considered the case of multiple kinases with
two states (figure 3a). In this case, multiple two-state kinases
in a futile cycle lead to multistationarity (figure 3a). With the
number of kinases n increasing, the number of steady states
linearly scales with n. We prove that the system can admit
at most 2n þ 1 steady states and further that n of them are
unstable (see the electronic supplementary material). The
other n þ 1 steady states are presumably stable. Secondly,
multistability can be achieved by one kinase with multiple
states (figure 3b). When the kinase has three distinct states,
the system can have three steady states at most, but a four-
state kinase results in the possibility of five steady states at
most (figure 3b; see the electronic supplementary material).
The general scenario with an n-state kinase is too complex
mathematically and does not admit the approach used to
analyse systems with multiple two-state kinases. However,
we make the conjecture that the number of positive steady
states grows linearly with n as well, such that the system
admits at most n þ 1 positive steady states if n is even and
n positive steady states if n is odd.
2.5. Multistability enables complex state transitionsThe above results confirm that a single futile signalling cycle
with a two-state kinase can generate bistable dynamics and
that such a system can be expanded by increasing the
number of kinase states to achieve unlimited multistationar-
ity. In this scenario, each additional kinase state potentially
drives the generation of a pair of steady states, one stable
and one unstable, due to the competition for the substrate.
Thus, it should be possible to use the total concentrations
(or kinetic parameters) of the different kinases to change
the signal thresholds to switch between steady states and
implement logic gates in this way. More specifically, in the
system with multiple two-state kinases, varying the total
concentration of a kinase can dictate the system transi-
tions among the different steady states resulting from
multistability.
Here, we show that by combinatorial perturbations of
different kinases, a system with three two-state kinases can
perform complex state transitions (figure 4). The varying par-
ameters are the total concentrations of the first two kinases,
namely [K1,tot] and [K2,tot]. We assume that the system
K1,r ↽⇀ K1,t K1 ↽
⇀ K2KnKn,r ↽
⇀ Kn,t, ... ,
↽⇀ , ... , ↽
⇀
S Sp S Sp
n = 2
n = 3 n = 4
n = 3
105
10.50
0.100.05
20 40 60 80 1 2 3 4 5
[Ktot] (mM)
[S]
(mM
)[S
] (m
M)
[S]
(mM
)[K2,tot] (mM)
50 100 150 200
[K1,tot] (mM)
10 20 30 40
[Ktot] (mM)
100
10
1
0.10
0.01
[S]
(mM
)
1000
10
0.100
0.001
1000
100
101
0.100
0.010
0.001
(a) (b)(i) (i)
(ii) (ii)
(iii) (iii)
Figure 3. Implementation of multistability by expanding the core bistable motif. (a) Multistability generated by a signalling cycle with multiple two-state kinases.(i) A schematic of the core bistable motif extended with multiple allosteric kinases. (ii) Bifurcation plot for a system with two allosteric kinases. The x-axis shows thesignal level [K2,tot] (total concentration of the second kinase K2), while the y-axis shows the steady-state level of [S] (unphosphorylated substrate). (iii) Bifurcationplot of a system with three allosteric kinases. The x- and y-axes are as above. Parameter values used for the bifurcation plots are listed in the electronic sup-plementary material, table S4. (b) Multistability generated by a signalling cycle with a multi-state kinase. (i) Schematic of a multi-state kinase catalysing afutile signalling cycle. (ii) Bifurcation plot of a system with a three-state kinase. (iii) Bifurcation plot of a system with a four-state kinase. The x- and y-axesare as above. In all bifurcation plots, solid lines correspond to stable steady states, while dashed lines correspond to unstable steady states. All axes use theunit of concentration micromolar. Parameter values used for the bifurcation plots are listed in the electronic supplementary material, table S5.
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starts off at a given state (O1 in figure 4) with low total con-
centration of both kinases. By increasing the total
concentration of either kinase (K1, K2) or both, the system
can be made to switch to three different end-states of [S]
(figure 4, points E3, E1 or E2). It is also possible to bring the
system into different states by perturbing the total concen-
trations of both kinases by a fixed amount each, but
following different sequential moves (figure 4, from O2 to
T1, T2, T3 and T4). In these examples, the final system
output is a function of the combinatorial activity patterns of
both kinases. By contrast, different perturbations would
result in the same output state in a monostable system. There-
fore, multistability can encode the specific order of changes
in the environmental signals (i.e. different kinase activities)
into different system outputs at steady state. The result is a
potential increase in the system’s capacity to store infor-
mation, e.g. relating to fluctuating or complex environments.
2.6. Natural signalling pathways display complexinteractions leading to multi-state enzymes andpotential for multistability
As discussed in the Introduction, futile signalling cycles are ubi-
quitous motifs in natural signalling networks, where they
feature multi-state enzymes. To demonstrate this point, we
explore two example cases of natural signalling cycles. The
first example comprises the signalling networks controlling
the cell cycle, in particular networks involving cyclin-depen-
dent kinases (Cdks). It is argued that the activity of Cdks is a
0
100
200
[K1,tot ] (mM
) [K 2,tot] (m
M)
300 150
100
50
E3
E2
E1
T4
T3
T2
T1O2
O1
log 10
[S]
log10[S]
0
–2
0
2 2
1
0
–1
–2
–3
Figure 4. Multistability installs complex state transitions. The steady-state level of the unphosphorylated substrate, [S], for different levels of the two kinases, [K1,tot]and [K2,tot]. The colour-coding shows the level of unphosphorylated substrate, [S], for each amount of kinase. The black and white dots represent specific states ofthe system. The black and white arrows show the hypothetical trajectories described when the kinase levels are perturbed in various combinatorial ways, as dis-cussed in the main text.
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key mechanism for ensuring appropriate switching dynamics
for the cell cycle [38–40]. The activity level of Cdk1 is regulated
by four different mechanisms: (i) activating phosphorylation by
Cdk-activating kinases (CAKs), where phosphorylation by a
CAK of a threonine residue increases the kinase activity of
Cdk1 [67]; (ii) inhibitory phosphorylation by Wee1, where
phosphorylation of a tyrosine residue by Wee1 reduces kinase
activity of Cdk1 [68]; (iii) cyclin binding, where cyclins binding
cooperatively to Cdk1 and their substrates promote Cdk1
kinase activity [69]; and (iv) Cdk-inhibitor (CKI) binding,
where CKIs bind to Cdk1 and block their active sites [68]
(figure 5a). Such combinatorial interactions (i.e. regulations)
where different Cdk1 ‘states’ (i.e. phosphorylated at different
positions, bound/unbound, etc.) compete for the same down-
stream substrates with potentially different activity levels, can
thus create a situation similar to that shown in figure 3b. It
must also be noted that there are multiple homologous Cdks
in the cell [41,68], which would replicate this situation of
multi-state enzymes competing for the same substrates to
result in a picture as shown in figure 3a. The system is further
complicated with the multi-state nature of the enzymes
upstream of Cdk’s. For example, Wee1 has differentially phos-
phorylated forms that can have different activity towards Cdk1
[70,71]. The second example for multi-state enzymes comes
from the MAPK signalling cascades [32]. For instance, the
MAPK signalling networks controlling yeast mating response
and filamentous growth response share the signalling proteins
Ste11 and Ste7, both of which have two phosphorylation sites
and can bind to a scaffolding protein Ste5 (figure 5b) [44].
The possible combinatorial interactions and the different
phospho-states of these proteins, as well as their downstream
interaction partners such as Fus3 and Kss1, provide a system
with multiple kinase states.
The picture emerging from the Cdk-based cell cycle as
well as the MAPK pathways is one with several multi-state
enzymes in competition for the same substrates. This picture
fits in the simplified motifs as analysed above, making it
theoretically possible for these pathways to display bistability
and multistability. For this possibility to be realized, the
different states of the key signalling enzymes in the natural
pathways must show different binding and catalytic activity
levels towards their substrates as mathematically analysed
above. The regulation of transitions among such states
would then be expected to form an important aspect of cellu-
lar dynamics and information processing. While this
viewpoint fits with some of the current knowledge, for
example, different Wee1 forms having different activity
levels, with some being indicated as ‘inactive’ [70,71], and
ubiquitination-based control of Wee1 being critical in cell
cycle progression [72], full experimental verification of the
relationship between cellular information processing and
the presented theory of multistability requires further studies.
A critical starting point would be measuring in vitro the
catalytic and binding rates of different enzyme forms found
in these systems to see if they fit with the mathematical
conditions for multistability presented here.
3. DiscussionThe key finding of this study is that the presence of a multi-
state kinase in the common futile signalling cycle motif
allows this functional interaction system to display bistability.
Thus, a phosphorylable substrate with a two-state kinase
forms one of the smallest bistable signalling motifs. The emer-
gence of bistability in this simple system relates closely to the
two states of the kinase forming two futile cycles that are com-
peting for the substrate. We define conditions on the kinetic
parameters of these two competing cycles that are necessary
and sufficient for three steady states. We show that these con-
ditions are met under biologically feasible parameter regimes.
Finally, we find that increasing either the number of two-state
CKI
CKI P
Cyc Cyc
(Cdcs, p53 etc.)mating responses
P
filamentousgrowth responses
filamentousgrowth signals
Tec1
Tec1
Fus3 Kss1
Ste7
Ste11
Ste7
Ste11
Tec1
Ste12
PP
PP
PP
PP
PP
PP
PP
P
Ste5
mating signals
Ste12 Ste12
Cyc
Cyc
P
Wee1
Cdc25 Cyc
Cdk1CAK
Cdk1
Cdk1
Cdk1 Cdk1
S Sp
(b)(a)
Figure 5. The bistable signalling motif in biological systems. (a) Different forms of regulation of Cdk1’s catalytic activity give rise to different states of Cdk1. Themultiple states of Cdk1 are involved in catalysing many downstream substrates, including Cdc and p53. Such catalytic reactions show precisely the structural patternin figure 3b. (b) The two MAPK cascades in yeast mating response and filamentous growth response. The two cascades share Ste11 and Ste7. Ste11, Ste7 and Kss1have two phosphorylation sites while Fus3 has three phosphorylation sites, one of which is phosphorylated by Ste5. This schematic shows that in both cascades allthree layers of signalling enzymes, MAP3 K (i.e. Ste11), MAP2 K (i.e. Ste7) and MAPKs (i.e. Fus3 and Kss1) exhibit different states that compete for their substrates.Thus, the cross-talk between the two cascades and the presence of the scaffold protein increases the number of states of the enzymes, resulting in a system similarto that considered in figure 3a.
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kinases acting on the same substrate or the number of distinct
states that a single kinase can exhibit increases the number of
steady states in an unbounded manner.
The core bistable signalling motif featuring multi-state
enzymes is prevalent in biological systems. The presence
of multiple conformational states with differential activity is
a common feature of many enzymes [57], and particularly
in signalling networks, where many kinases and phospha-
tases admit multiple states that display different levels of
activity and that are regulated through covalent modification
or interaction with scaffold proteins [42,73]. As we have
shown above, using Cdks and MAPK pathways as examples,
there are several natural cases where such interactions create
or embed the described core bistable motifs or extensions
of it. Our findings thus provide mathematical proof that
these natural systems can theoretically allow bistability and
potentially unbounded multistability. Transitions between
the steady states can underpin the capacity of cells to
map environmental states to internal gene expression and
physiology, increasing their ability to adapt to different
or fluctuating environments. The validation and further
interrogation of these possibilities must come from exper-
imental studies. In particular, synthetic biology approaches
can be used to implement the core bistable motif described
here using existing multi-state proteins and kinases from
nature and analysing their dynamics in a controlled manner.
These approaches are already being employed to study
MAPK and two-component signalling systems [73–76], and
can be further extended using the presented results as guiding
principles for experiments.
An intuitive interpretation of our results is that compe-
tition of different futile cycles for the same substrate is a
key prerequisite for bistability. This intuitive view can also
be applied to understand previously described bistable and
multistable signalling motifs. For instance, a substrate with
multiple phosphorylation sites that are acted upon by the
same kinase is shown to implement bistability and multi-
stability [31–34,77]. This system is almost a symmetric
version of the system we consider here, as it features futile
cycles involving differently phosphorylated substrates com-
peting for the same enzyme. Another example of a bistable
system is where a futile cycle can take place in two different
compartments, with both substrates and enzymes shuttling
between the two compartments. This again fits our intuitive
view, where the separation of enzymes and substrates in
different compartments creates a set of futile cycles that are
competing for both substrates and enzymes [54].
These examples indicate that competing futile cycles
could provide a general condition for determining bistability.
In order to validate this idea, further exploration of different
motifs and the structural conditions on multistationarity is
required. One possible approach would be to enumerate a
large set of small signalling networks and compare structural
differences between monostationary and multistationary net-
works. Specific structural patterns might emerge, which can
be validated by further mathematical analyses. These math-
ematically derived conditions can then be used to better
understand natural signalling systems and design bistable
signalling networks and biochemical memory through
synthetic biology.
4. Material and methods4.1. Core model for a futile signalling cycle with a two-
state kinaseThe core futile signalling cycle we consider here consists of a
covalent modification, i.e. de/phosphorylation, of a substrate by
a kinase and a phosphatase [9,78]. We extend this motif by consid-
ering multiple possible states of the kinase that could potentially
have differing activity levels. For the case of the two-state kinase,
we consider two distinct states (relaxed, Kr, and tense, Kt) catalysing
a substrate (S) into a phosphorylated product (Sp). To simplify the
system, we do not model the phosphatase directly, but rather con-
sider an auto-dephosphorylation reaction. The set of reactions we
consider include the transformations between the different kinase
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states and Michaelis–Menten mechanisms for the phosphorylation
of the substrate:
Kr þ S Ok1
k2
KrS�!k3 Kr þ Sp,
Kt þ S Ok4
k5
KtS�!k6 Kt þ Sp,
Sp�!k7 S,
Kr Ok8
k9
Kt
and KrS Ok10
k11
KtS:
9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;
ð4:1Þ
The parameters k1 to k11 represent the kinetic parameters, also
called rate constants. The system consists of six species, two of
which are substrate–enzyme complexes. Assuming the law
of mass action, we model the rate of change of the concentra-
tions of each of the species with the following system of ordinary
differential equations:
d[Kr]
dt¼�k1[Kr][S]þk2[KrS]þk3[KrS]�k8[Kr]þk9[Kt],
d[Kt]
dt¼�k4[Kt][S]þk5[KtS]þk6[KtS]þk8[Kr]�k9[Kt],
d[KrS]
dt¼ k1[Kr][S]�k2[KrS]�k3[KrS]�k10[KrS]þk11[KtS],
d[KtS]
dt¼ k4[Kt][S]�k5[KtS]�k6[KtS]þk10[KrS]�k11[KtS],
d[S]
dt¼�k1[Kr][S]þk2[KrS]�k4[Kt][S]þk5[KtS]þk7[Sp]
andd[Sp]
dt¼ k3[KrS]þk6[KtS]�k7[Sp]:
9>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>;
ð4:2Þ
The system satisfies the following two conservation laws:
[Ktot] ¼ [Kr]þ [Kt]þ [KrS]þ [KtS]
and [Stot] ¼ [S]þ [Sp]þ [KrS]þ [KtS],
)ð4:3Þ
where we introduce two concentration invariants, namely [Ktot]
and [Stot], representing the total concentration of the kinase
and the substrate, respectively. These equations arise from
the biological assumption, implicit in the choice of reactions,
that the total concentrations of the signalling proteins are
constant over the relevant timescales of signalling (i.e. the
model does not consider dynamics arising from gene regulation
and expression).
4.2. Mathematical analysis of the steady states of thecore motif
We consider the steady-state equations (obtained by setting the
differential equations in equation (4.2) to zero) and the conserva-
tion law for [Ktot] to express the steady-state concentrations of all
variables in terms of the concentration of the unphosphorylated
substrate (see the electronic supplementary material). Using the
conservation law for [Stot], we obtain then a polynomial,
whose positive solutions are the steady-state levels of [S] (see
the electronic supplementary material, equation (9)). The coeffi-
cients of the polynomial depend on the rate constants and the
total concentrations. We show that the polynomial has one,
two or three positive roots (depending on the parameters),
which proves that the core motif admits three steady states for
a proper choice of rate constants and total concentrations of Sand K ([Stot] and [Ktot], respectively). We further show that one
of the three steady states is unstable. Simulations show that the
other two steady states are always stable.
We can further derive analytical conditions for the poly-
nomial to have three positive roots, resulting in necessary (but
not sufficient) conditions for bistability that depend on both
the total concentrations and the rate constants:
a1Ktot þ a2 , Stot , a3Ktot þ a4, ð4:4Þ
where a1, a2, a3 and a4 are positive expressions in the rate con-
stants (see full expression in the electronic supplementary
material, equation(10)).
Following an alternative approach, based on Brouwer Degree
Theory (see [79,80], where the strategy was applied to study a
multi-site phosphorylation system), we derive sufficient and
necessary conditions on the 11 rate constants alone, as discussed
in the Results, under which the system has three steady states
(for the complete proof, see the electronic supplementary
material, proposition 4). Namely:
(k3 � k6)(hrk9k10 � htk8k11) . ((k6 þ k7)k10 þ (k3 þ k7)k11)
(hrk10 þ htk11), ð4:5Þ
where hr ¼ k1=ðk2 þ k3Þ and ht ¼ k4=ðk5 þ k6Þ are the inverse of
the Michaelis–Menten constants of the two conformational states
of the kinase. While equation (4.5) presents the sufficient and
necessary conditions on the 11 rate constants alone, we show
in the electronic supplementary material, that if the parameters
satisfy this equation, then there exists total concentrations [Stot]
and [Ktot] that result in three steady states in the system (see
the electronic supplementary material, proposition 6).
Necessary and sufficient conditions for three steady states
involving both the rate constants and the total concentrations
can also be found for this system (see the electronic supplementary
material, proposition 6). The expressions we obtain are complex
and do not allow for easy biological interpretation. A guide on
how to generate parameters that yield to three steady states is
given in the electronic supplementary material (section 1.7).
4.3. Proof of unbounded multistabilityThe results for the extended models with n two-state kinases
competing for the same substrate, or a single kinase with mul-
tiple states are detailed in the electronic supplementary
material, sections 2 and 3. There, we show that there are rate con-
stants for which the system with n allosteric kinases has exactly
2m þ 1 (where m ¼ 0, . . . ,n) positive steady states. This holds
also for m ¼ n, in which case there are 2n þ 1 positive steady
states. In fact, we show that 2n þ 1 is the maximal possible
number of steady states, positive as well as steady states
for which at least one concentration is zero. To show these
results, we follow the model reduction techniques from [81,82].
These techniques enable us to conclude the existence of
multiple steady states for the networks of interest, from their
existence for simpler networks that allow for a detailed
mathematical analysis.
4.4. Checking bistabilityWe used the Chemical Reaction Network Theory Toolbox (CRNT
Toolbox 2.3, https://crnt.osu.edu/CRNTWin) to analyse the net-
works shown in figure 1 and the electronic supplementary
material, figure S1. The toolbox enables us to determine whether
a (bio)chemical reaction network can have at least two steady
states (for some choice of parameters) solely based on the struc-
ture of the network, assuming all reactions follow mass-action
law. For each of the networks in figure 1 and electronic sup-
plementary material, figure S1, we obtained a set of parameters
for which the network admits at least two steady states. We
then checked that, in each case, there were actually three
steady states, two of which were stable and one unstable. For
an introduction to the toolbox, we refer the reader to the
manual that is downloaded with the software.
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4.5. Parameter samplingTo sample from the parameter space of the rate constants, we
assume ln ki follows a uniform distribution, ln ki � U(a, b), (i ¼1, . . . ,11). We use a = ln 1023 and b = ln 103, based on biologically
feasible parameters validated in experiments [83] (electronic
supplementary material, table S1).
To sample a parameter set without the thermodynamic con-
straint, we simulate 11 uniform variables ui, i ¼ 1, . . . ,11 with
common distribution U(0, 1) and define ki ¼ e(b2 a)uiþ a. To
sample rate constants with the thermodynamic constraint, we use
a different approach. Under the constraint, two products of four kin-
etic rates are identical k1k5k9k10¼ k2k4k8k11, which translates into
an equality of two sums of logarithmic rates. We first assume the
logarithmic rates lie between 0 and 1, and then afterwards transform
them such that they lie between a and b. We follow three steps.
Step 1. In this step, we simulate the sum of the logarithmic
rates using rejection sampling. To simulate the sum, we simulate
four uniform variables u1, u2, u3, u4, with distribution U(0,1) and
define s = u1 + u2 + u3 + u4. To take into account the constraint, the
value s is accepted with the probability p(s) that the sum of the
other four logarithmic rates takes the same value. This prob-
ability is derived from the Irwin–Hall distribution [84,85], and
takes the form:
pðsÞ ¼
14s
3 if 0 � s � 114ð�3s3 þ 12s2 � 12sþ 4Þ if 1 , s � 214ð3s3 � 24s2 þ 60s� 44Þ if 2 , s � 314ð�s3 þ 12s2 � 48sþ 64Þ if 3 , s � 4:
8>>>><>>>>:
ð4:6Þ
The probability might attain any value between 0 and
1. A random number z is then drawn from a uniform variable
Z � U(0,1). If z , p(s), we accept s. If not, we discard s, redraw
ui, z and recalculate p(s) until s is accepted. We do this as
many times as we want parameter sets.
Step 2. In this step, we simulate the individual logarithmic
rates that sum up to the accepted values from the first step. For
each accepted s, we simulate four uniform variables U(0, 1).
Denote the variables by pi, i ¼ 1, . . . ,4, and let their sum be P.Define v1 ¼ s � p1=P, v5 ¼ s � p2=P, v9 ¼ s � p3=P, v10 ¼ s � p4=P,
such that v1, v5, v9 and v10 add up to s. If v1, v5, v9, v10 are all
less than one, accept them. Otherwise repeat the procedure by
redrawing four new uniform variables until the condition is ful-
filled. Next we do the same to obtain v2 ¼ s � q1=Q, v4 ¼ s � q2=Q,
v8 ¼ s � q3=Q, v11 ¼ s � q4=Q, where qi, i ¼ 1, . . . ,4, are uniform
U(0,1) random variables with sum Q.
Step 3. We generate three random numbers v3, v6 and v7 from
a uniform distribution U(0, 1). Finally, we compute the rate
constants as ki ¼ e(b�a)viþa, i ¼ 1, . . . ,11.
Note that without the thermodynamic constraint, all rate con-
stants are generated from the same distribution. With the
constraint, k3, k6 and k7 follow the same distribution as without
the constraint. The remaining eight parameters follow a different
distribution, as they are constrained. This distribution is common
to all eight parameters as the equation for the constraint is
symmetric in these parameters.
Competing interests. We declare we have no competing interests.
Funding. M.S., C.W. and E.F. acknowledge funding from the LundbeckFoundation (Denmark). C.W. and E.F. acknowledge funding from theDanish Research Council. O.S.S. and S.F. acknowledge support fromUniversity of Warwick, School of Life Sciences and funding fromEngineering and Physical Sciences Research Council, grant no. EP/H04986X/1.
Acknowledgements. We thank Mart Loog for insightful comments on anearlier version of this manuscript.
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